4.2 Congruence and 4.2 Congruence and TrianglesTriangles

Congruent FiguresCongruent Figures

• 2 figures are congruent 2 figures are congruent if they have the exact if they have the exact same size and shape.same size and shape.

• When 2 figures are When 2 figures are congruent the congruent the corresponding parts corresponding parts are congruent. (angles are congruent. (angles and sides)and sides)

• Quad ABDC is Quad ABDC is congruent to Quad congruent to Quad EFHGEFHG

AB

C D

EF

GH

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___))))))) (

((

())))

)))((

ZZ

•If If ΔΔ ABC is ABC is to to ΔΔ XYZ, which angle is XYZ, which angle is to to C?C?

Thm 4.3Thm 4.333rdrd angles thm angles thm

• If 2 If 2 s of one s of one ΔΔ are are to 2 to 2 s of another s of another ΔΔ, then the , then the 3rd 3rd s are also s are also ..

Ex: find x

)

))

2222oo

8787oo

)

)) (4x+15)(4x+15)oo

Ex: continuedEx: continued

22+87+4x+15=18022+87+4x+15=180

4x+15=714x+15=71

4x=564x=56

x=14x=14

Ex: ABCD is to HGFE, find x and y.AB

DD CC

F E

GGHH

9191oo

8686oo

9cm

(5y-12)(5y-12)oo

4x-3cm4x-3cm

113113oo

4x-3=9 5y-12=113

4x=12 5y=125

x=3 y=25

Thm 4.4Props. of Δs

• Reflexive prop of Δ - Every Δ is to itself (ΔABC ΔABC).

• Symmetric prop of Δ - If ΔABC ΔPQR, then ΔPQR ΔABC.

• Transitive prop of Δ - If ΔABC ΔPQR & ΔPQR ΔXYZ, then ΔABC ΔXYZ.

A

BB

CPP

Q

R

XY

Z

GivenGiven: seg RP : seg RP seg MN, seg PQ seg MN, seg PQ seg NQ , seg NQ , seg RQ seg RQ seg MQ, m seg MQ, mP=92P=92oo and m and mN is 92N is 92oo..

ProveProve: : ΔΔRQP RQP ΔΔMQNMQN

R

P

Q

N

M

9292oo

9292oo

Statements ReasonsStatements Reasons1. 1. given1. 1. given2. m2. mP=mP=mN 2. subst. prop =N 2. subst. prop =3. 3. P P N N 3. def of 3. def of s s 4. 4. RQP RQP MQN 4. vert MQN 4. vert s thms thm5. 5. R R M 5. 3M 5. 3rdrd s thms thm6. 6. ΔΔRQP RQP ΔΔ MQN 6. def of MQN 6. def of ΔΔss